I've a similar problem of certain small button or icon-type images not loading. Am using TTF 38.10.0 on a Mac G4 10.4.11 An old work-horse, yes, but I'd like to continue using it. The problem is seen in the attached image, where small tall rectangles with appear instead of images. They have the alphanumeric E9 or similar in the top half, and another number in the bottom half. I use add-ons Ghostery 5.4.10, self-destructing cookies 0.4.9, adblocker ultimate 2.26 - but whitelisting a site doesn't help.
James, please open a separate ticket for your issue, since this is not related to the removal of the Add-ons tab. Also, a URL so we can see if this reproduces on current versions of TenFourFox would be helpful.
To the other users, you can still download older versions of addons and install them by simply dropping them on any open browser window. However, the "Get Add-ons" tab is controlled by Mozilla, not us, and offers addon versions we aren't compatible with anymore (which is why it was removed). You can browse for these older addons from https://addons.mozilla.org/, searching for the addon you want, and looking for any version compatible with Firefox 45 under "See all versions."